Isospin breaking quark condensates in Chiral Perturbation Theory

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Isospin-breaking quark condensates in chiral perturbation theory

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IOP PUBLISHING JOURNAL OF PHYSICS G: NUCLEAR AND PARTICLE PHYSICS

J. Phys. G: Nucl. Part. Phys. 39 (2012) 015004 (21pp) doi:10.1088/0954-3899/39/1/015004

Isospin-breaking quark condensates in chiralperturbation theory

A Gomez Nicola and R Torres Andres

Departamento de Fsica Teorica II, Univ. Complutense. 28040 Madrid, Spain

E-mail: [email protected] and [email protected]

Received 30 July 2011

Published 1 December 2011

Online at stacks.iop.org/JPhysG/39/015004

AbstractWe analyze the isospin-breaking corrections to quark condensates within one-

loop SU(2) and SU(3) chiral perturbation theory including mu = md as wellas electromagnetic (EM) contributions. The explicit expressions are given and

several phenomenological aspects are studied. We analyze the sensitivity of

recent condensate determinations to the EM low-energy constants (LEC).

If the explicit chiral symmetry breaking induced by EM terms generates a

ferromagnetic-like response of the vacuum, as in the case of quark masses, the

increasing of the order parameter implies constraints for the EM LEC, which

we check with different estimates in the literature. In addition, we extend the

sum rule relating quark condensate ratios in SU(3) to include EM corrections,

which are of the same order as the mu = md ones, and we use that sum ruleto estimate the vacuum asymmetry within ChPT. We also discuss the matchingconditions between the SU(2) and SU(3) LEC involved in the condensates,

when both isospin-breaking sources are taken into account.

(Some figures may appear in colour only in the online journal)

1. Introduction

The low-energy sector of QCD has been successfully described over recent years within the

chiral Lagrangian framework. Chiral perturbation theory (ChPT) is based on the spontaneous

breaking of the chiral symmetry SUL(Nf) SUR(Nf) SUV(Nf) with Nf = 2, 3 lightflavours and provides a consistent and systematic model-independent scheme to calculate

low-energy observables [13]. The effective ChPT Lagrangian is constructed as the moregeneral expansion L = Lp2 + Lp4 + compatible with the QCD underlying symmetries,where p denotes derivatives or meson mass and external momentum below the chiral scale

1 GeV.The SUV(Nf) group of vector transformations corresponds to the isospin symmetry for

Nf = 2. In the Nf = 3 case, the vector group symmetry is broken by the strange-light quarkmass difference ms mu,d, although ms can still be considered as a perturbation compared to , leading to SU(3) ChPT [3]. In the Nf = 2 case, the isospin symmetric limit is a very good0954-3899/12/015004+21$33.00 2012 IOP Publishing Ltd Printed in the UK & the USA 1
http://dx.doi.org/10.1088/0954-3899/39/1/015004mailto:[email protected]:[email protected]://stacks.iop.org/JPhysG/39/015004http://stacks.iop.org/JPhysG/39/015004mailto:[email protected]:[email protected]://dx.doi.org/10.1088/0954-3899/39/1/015004


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J. Phys. G: Nucl. Part. Phys. 39 (2012) 015004 A Gomez Nicola and R Torres Andres

approximation in nature. However, there are several known examples where isospin breaking

is phenomenologically relevant at low energies, such as sum rules for quark condensates [3],

meson masses and corrections to Dashens theorem [4], pionpion [5, 6] and pionkaon [7, 8]

scattering in connection with mesonic atoms [9, 10], CP violation [11], a0

f0 mixing [12],

kaon decays [13, 14] and other hadronic observables (see [15] for a recent review).The two possible sources of isospin breaking are the md mu light quark mass

difference and electromagnetic (EM) interactions. Both can be accommodated within the

ChPT framework. The former is accounted for by modifying the quark mass matrix and

generates a 0 mixing term in the SU(3) Lagrangian [3]. The expected corrections from

this source are of order (md mu)/ms. On the other hand, EM interactions, which inparticular induce mass differences between charged and neutral light mesons, can be included

in ChPT via the external source method and give rise to new terms in the effective Lagrangian

[46, 1618] of order Le2 , Le2p2 and so on, with e the electric charge. These terms fit into

the ChPT power counting scheme by considering formally e2 = O(p2/F2), with F the piondecay constant in the chiral limit.

Thepurpose of this paper is to study the isospin-breaking corrections to quark condensates,

whose main importance is their relation to the symmetry properties of the QCD vacuum.The singlet contributions uu + dd for SU(2) and uu + dd + ss for SU(3) are orderparameters for chiral symmetry, while the isovector one uu dd behaves as an orderparameter for isospin breaking, which is not spontaneously broken [ 19]. We will calculate the

condensates within one-loop ChPT, which ensures the model independence of our results, and

will address several phenomenological consequences. The two sources of isospin breaking

will be treated consistently on the same footing, which will allow us to test the sensibility

of previous phenomenological analysis to the EM low-energy constants (LEC). Moreover,

the EM corrections induce an explicit breaking of chiral symmetry which will lead to lower

bounds for certain combinations of the LEC involved, provided the vacuum response is

ferromagnetic, as in the case of quark masses. In addition, in SU(3) one can derive a sum

rule relating the different condensate ratios for mu = md [3] which, as we will show here,receives an EM correction not considered before and of the same order as that proportional

to mu md. The latter is useful to estimate the vacuum asymmetry dd/uu reliably withinChPT. An additional aspect that we will discuss is the matching of the SU(2) and SU(3) LEC

combinations appearing in the condensates when both isospin-breaking sources are present,

comparing with previous results in the literature. The analysis carried out in this work will

serve also to establish a firm phenomenological basis for its extension to finite temperature, in

order to study different aspects related to chiral symmetry restoration [20].

With the above motivations in mind, the paper is organized as follows: in section 2 we

briefly review the effective Lagrangian formalism needed for our present work, paying special

attention to several theoretical issues and to the numerical values of the parameters and LEC

needed here. Quark condensates for SU(2) are calculated and analyzed in section 3, where

we discuss the general aspects of the bounds for the EM LEC based on chiral symmetry

breaking. In that section we also comment on the analogy with lattice analysis. The SU(3)

case is separately studied in section 4. In that section, we first perform a numerical analysisof the isospin-breaking corrections, paying special attention to the effect of the EM LEC in

connection with previous results in the literature. In addition, we obtain the EM corrections

to the sum rule for condensate ratios, which we use to estimate the vacuum asymmetry

within ChPT. We also provide the LEC bounds for this case, checking them with previous

LEC estimates and, finally, we discuss the matching conditions for the LEC involved. In the

appendix we collect the Lagrangians of fourth order and the renormalization of the LEC used

in the main text.

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2. Formalism: effective Lagrangians for isospin breaking

The effective chiral Lagrangian up to fourth order is given schematically by

Leff

=Lp2

+e2

+Lp4

+e2p2

+e4 . (1)

The second-order Lagrangian is the familiar nonlinear sigma model, including now the gauge

coupling of mesons to the EM field through the covariant derivative, plus an extra term:

Lp2+e2 =F2

4tr [DU

DU + 2B0M(U+ U)+ Ctr[QUQU]. (2)

Here, F is the pion decay constant in the chiral limit and U(x) SU(Nf) is the Goldstoneboson (GB) field in the exponential representation U = exp[i/F] with

SU(2) : =0

2+

2 0,

SU(3) :

=

0 + 1

3

2+

2K+

2

0

+1

3

2K0

2K 2K0 23

, (3)

with the octet member with I3 = S = 0. The covariant derivative is D = + iA[Q, ]with A the EM field. M and Q are the quark mass and charge matrices, respectively, i.e. in

SU(3)M = diag(mu,md,ms) and Q = diag(eu, ed, es) with eu = 2e/3, ed = es = e/3 forphysical quarks. The additional term in (2), the one proportional to C, can be understood as

follows: the QCD Lagrangian for mu = md coupled to the EM field is not invariant under anisospin transformation q gq with g SU(Nf) and q the quark field. However, it would beisospin invariant if the quark matrix Q is treated as an external field transforming as Q gQg.Therefore, the low-energy effective Lagrangian has to include all possible terms compatible

with this new symmetry, in addition to the standard QCD symmetries. The lowest orderO(e2)

is the C-term in (2), since U transforms as U gUg. Actually, one allows for independentspurion fields QL(x) and QR(x) transforming under SUL(Nf)

SUR(Nf) so that one can

build up the new possible terms to any order in the chiral Lagrangian expansion, taking in theend QL = QR = Q [4].

In the previous expressions, F,B0mu,d,s,C are the low-energy parameters to this order.

Working out the kinetic terms, they can be directly related to the leading-order (LO) tree-level

values for the decay constants and masses of the GB. In SU(2), the tree-level masses to LO

are

M2+ = M2 = 2mB0 + 2Ce2

F2,

M20 = 2mB0, (4)with m = (mu + md)/2 the average light quark mass. Note that both terms contributing to thecharged pion mass are of the same order in the chiral power counting, although numerically

M2 M2

0 /M2

0

0.1, which we will use in practice as a further perturbative parameter

to simplify some of the results.

In the SU(3) case, the mass term in (2) induces a mixing contribution between the 0 and

the meson fields given by Lmix = (B0/

3)(md mu)0. Therefore, the kinetic term hasto be brought to the canonical form before identifying the GB masses, which is performed by

the field rotation [3]:

0 = 0 cos sin , = 0 sin + cos , (5)

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where the mixing angle is given by

tan2 =

3

2

md mums m

. (6)

Once the above 0

rotation is carried out, the SU(3) tree-level meson masses to LO read

M2+ = M2 = 2mB0 + 2Ce2

F2,

M20 = 2B0

m 23(ms m)

sin2

cos2

,

M2K+ = M2K = (ms + mu)B0 + 2Ce2

F2, (7)

M2K0

= (ms + md)B0,

M2 = 2B0

1

3(m + 2ms)+

2

3(ms m)

sin2

cos2

.

The above five equations are the extension of the Gell-MannOakesRenner (GOR)relations [21] to the isospin asymmetric case and allow us to relate the four constants B0mu,d,sand C ( is given in terms of quark masses in (6)) with the five meson masses or their

combinations. The additional equation provides the following relation between the tree-level

LO masses: M2K M2

2 3M2 M2K0M2K0 M20 = 0. (8)The above equation is compatible with the one obtained in [22] neglectingO(mu md)2 terms.Actually, note that although all terms in (7) are formally of the same chiral order, numerically

(see below) we expect (3/4)(mdmu)/ms 1 and hence the mixing-angle correctionsto the squared masses to be O(M2) and O(M

2

2) for the neutral pion and eta, respectively.

On the other hand, in the isospin-symmetric limit (mu = md and e = 0), (8) is nothing but theGell-MannOkubo formula 4M2K

3M2

M2

=0 [23]. Neglecting only the md

mu mass

difference in (7) leads to Dashens theorem M2K M2K0 = M2 M20 [24] and then equation(8) reduces to 4M2

K0 3M2 M20 = 0, i.e. the Gell-MannOkubo formula for neutral states.

However, the violation of Dashens theorem at tree level due to those quark mass differences

is significant numerically for kaons. In our present treatment we consider those differences

on the same footing as the EM corrections to the masses. For pions, the main effect in the

0 + mass difference comes from the EM contribution [25].All the previous expressions hold for tree-level LO masses M2a with a = , 0,K, ,

in terms of which we will write all of our results. They coincide with the physical masses

to LO in ChPT, i.e. M2a,phys = M2a(1 +O(M2)). Calculating the ChPT corrections to a givenorder then allows us to determine the numerical values of the tree-level masses, knowing their

physical values and to that order of approximation. The same holds for F, which coincides

with the meson decay constants in the chiral limit F2a,phys = F2(1 +O(M2)). Next to leading

order (NLO)O(M2

) corrections to meson masses and decay constants were given in [2, 3] fore2 = 0. EM corrections to the masses can be found in [4] for SU(3) and in [6, 18] for SU(2)including both e2 = 0 and mu = md isospin-breaking terms.

The fourth-order Lagrangian in (1) consists of all possible terms compatible with the

QCD symmetries to that order, including the EM ones. The Lp4 Lagrangian is given in [2] for

the SU(2) case, h1,2,3 (contact terms) and l1...7 denoting the dimensionless LEC multiplying

each independent term, and in [3] for SU(3) the LEC named H1,2 and L1...10. The EM Le2p2

and Le4 for SU(2) are given in [5, 6], k1,...13 denoting the corresponding EM LEC, and in [4]

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for SU(3) with the K1...17 EM LEC. For completeness, in the appendix we give the relevant

terms needed in this work. The LEC are renormalized in such a way that they absorb all the

one-loop ultraviolet divergences coming from Lp2 and Le2 , according to the ChPT counting,

and depend on the MS low-energy renormalization scale in such a way that the physical

quantities are finite and scale independent. The renormalization conditions for all the LEC canbe found in [2, 4, 6, 17] and in the appendix we collect only those needed in this work.

As customary, we denote the scale-dependent and renormalized LEC by a superscript r.

The renormalized LEC are independent of the quark masses by definition, although their finite

parts are unknown, i.e. they are not provided within the low-energy theory. The numerical

values of the LEC at a given scale can be estimated by fitting meson experimental data,

theoretically by matching the underlying theory under some approximations, or from the

lattice. These procedures allow us to obtain estimates for the real-world LEC at the expense

of introducing residual dependences of those LEC on the parameters of the approximation

procedure, which typically involves a truncation of some kind. Examples of these are the msdependence on the SU(2) LEC when matching the SU(3) ones, the correlations between LEC,

masses and decay constants through the fitting procedure, the QCD renormalization scale and

gauge dependence of some of the EM LEC or the dependence with lattice artifacts such asfinite size or spurious meson masses. We will give more details below, specially regarding the

EM LEC which will play an important role in our present work. An exception to the LEC

estimates are the contact LEC hi and Hi, which are needed for renormalization but cannot

be directly measured. The physical quantities depending on them are therefore ambiguous,

which comes from the definition of the condensates in QCD perturbation theory, requiring

subtractions to converge [2]. It is therefore phenomenologically convenient to define suitable

combinations which are independent of the hi,Hi. We will bear this in mind throughout this

work, providing such combinations when isospin breaking is included.

We will analyze in one-loop ChPT (NLO) the quark condensates, which for a given flavour

qi can be written at that order as

qiqi

= Leff

mi . (9)

The above equation is nothing but the functional derivative with respect to the ith

component of the scalar current, particularized to the values of the physical quark masses,

according to the external source method [2, 3]. Therefore, we will be interested only in the

terms of the fourth-order Lagrangian containing at least one power of the quark masses.

These are the operators given in equations (A.1) and (A.2) for SU(2) and SU(3), respectively.

Thus, the LEC that enter our calculation are l3, h1, h3, k5, k6, k7 in SU(2), and L6,L8,H2,

K7,K8,K9,K10 in SU(3). Besides, up to NLO, only tree-level diagrams from the fourth-order

Lagrangian can contribute to the condensates, so that in practice it is enough to set U = 1 in(A.1)(A.2) for getting those tree-level contributions from (9).

2.1. Masses and LECs

For most of the numerical values of the different LECs and parameters in the SU(3) case, we

will follow [26], where fits to Kl4 experimental data are performed in terms ofO(p6) ChPT

expressions,including the isospin massdifference mu/md = 1 andEM corrections to themesonmasses, extending a previous work [27] where isospin breaking was not considered. Those fits

have been improved in a recent work [28], which takes into account new phenomenological

and lattice results. We will however stick to the values of [26], since our main interest is to

compare the isospin-breaking condensates with the two sources included and to estimate the

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effect of the EM LEC. In the new fits [ 28] isospin breaking is included only to correct for

the charged kaon mass and the condensate values are not provided. For a review of different

estimates of the quark masses and condensates see also [29] and [30]. In addition, in [31] a

recent update of lattice results for low-energy parameters can be found, including LEC and

the quark condensate. We will use the central values of the main fit in [26]. The value ofms/m = 24 [29, 30] is used as an input in [26], as well as Lr6 = 0, as follows e.g. from OZI ruleor large-Nc arguments [3]. The more recent fits [28] consider an updated value ofms/m = 27.8in accordance with recent determinations [31] and a nonzero value of Lr6 is obtained as an

output. The suppression ofLr6 has been questioned in connection with a reduction of the light

quark condensate when the number of flavours is increased [32], within the framework of

generalized ChPT. In that context, the chiral power counting is modified due to the smallness

of the condensate. Here, we will adhere to the standard ChPT picture, where the condensate

and the GOR-like relations are dominated by the LO [33], sustained by the recent lattice LEC

estimates [31]. The values of F = 87.1 MeV, 2B0m = 0.0136 GeV2, mu/md = 0.46 andL8( = 770 MeV) = 0.62 103 are outputs from the main fit in [26]. With those values weobtain from (6) = 0.014 and from (7) the tree-level masses of0,K0, .

To calculate the tree-level charged meson masses, we also need the value of theCconstant,which can be inferred also from the results in [26] since the EM correction is numerically very

small in the charged kaon mass with respect to the pure QCD contribution. This allows us to

extract the tree-level charged kaon mass directly from the expressions for MK/MK,QCD in

[26], approximating MK,QCD by the full physical mass. From there we extract the value ofC

by subtracting the tree-level QCD part in (7) calculated with the above given quark masses.

In this way we obtain C = 5.84 107 MeV4, which is very close to the values obtainedsimply from the chargedneutral pion mass difference in (7) setting the masses and F to their

physical values [6] or from resonance saturation arguments [4]. From that C value we obtain

the tree-level charged pion mass, using again (7). Nevertheless, to the order we are calculating

we could have used the physical meson masses and decay constants instead of the tree level

ones as well, since formally the difference is hidden in higher orders. The main reason why we

choose the values in [26] is to compare directly with their numerical quark condensates and

estimate the importance of the Kri corrections (see section 4.1 for details). The constant H2 willalso appear explicitly in quark condensates. Since it cannot be fixed with meson experimental

data, when needed we will estimate it from scalar resonance saturation arguments asHr2 = 2Lr8[16, 26], although we will comment below more about the Hr2 dependence of the results and

provide physical quantities which are independent of the contact terms.

Regarding the EM LEC, the SU(3) Kri have been estimated in the literature under different

theoretical schemes. Resonance saturation was used in [35], large-Nc and NJL models in [34],

complemented with QCD perturbative information in [36] and a sum-rule approach combined

with low-lying resonance saturation has been followed in [37, 38]. The works [34, 3638]

have in common the use of perturbative QCD methods for the short-distance part of the

LEC and different model approaches for the long-distance part. This procedure implies that

the LEC estimated in that way depend (roughly logarithmically) in general on the QCD

renormalization scale, which we call 0 to distinguish it from the low-energy scale , aswell as on the gauge parameter. A closely related problem is that the separation of the

strong (e = 0) and EM contributions for a given physical quantity is in principle ambiguous[34, 37, 39, 40]. The origin of this ambiguity [40] is that QCD scaling quantities such as quark

masses also contain EM contributions through the renormalization group evolution in the full

QCD+EM theory. Thus, a particular prescription for disentangling those contributions must be

provided. In addition, when matching such quantities between the low-energy sector and the

underlying theory, the choice of a given prescription will necessarily affect the scale and gauge

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dependence of the EM LEC. These theoretical uncertainties, as well numerical ones, make

those theoretical EM LEC estimates not fully compatible. For these reasons, in many works

analyzing EM corrections, the LEC are simply assumed to lie within natural values |Kri |, |kri | 0 [47].

Our purpose here will be to explore the consequences of that EM ferromagnetic behaviorto LO. If the vacuum response is ferromagnetic, certain bounds for the EM LEC involved

should be satisfied. We will derive those bounds and show that they are independent of the

low-energy scale and thus can be checked in terms of physical quantities. Next we will check

that the bounds are satisfied for the different estimates available for the EM LEC, with more

detail for the SU(3) case in section 4.4, where we also discuss the gauge independence of our

results. This will provide a consistency check for the ferromagnetic behavior.

An important comment is that we will discuss the ferromagnetic-like condition on the

EM correction to qq and, as explained above, the splitting of the e = 0 and e = 0 parts inQCD+EM is ambiguous [39, 40]. This does not affect the low-energy representation of the

condensates, which can be written in terms of physical quantities such as meson masses and

decay constants. However, we will test our bounds with the EM LEC estimates obtained by

matching low-energy results with the underlying theory [3437]. Therefore, we have to beconsistent with the prescription for charge splitting followed in those works. This amounts

to the direct separation of the e = 0 part, which still may contain residual charge and 0QCD scale dependence through running parameters. The consequence is that the EM LEC

thus defined are in general 0-dependent, as discussed in [40]. Therefore, those estimates are

reliable only if there is a stability range where the dependence on 0 is smooth and lies within

the theoretical errors [34]. Actually, such stability range criteria are met for the LEC involved

in our analysis (see section 4.4). Within that range, our identification of the e2-dependent part

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in qq is consistent with the splitting scheme followed in those works. Actually, in that schemeF0 is 0-independent and the e

2-dependent part of the 0 running ofB0mu,d is the same as that

ofmu,d in perturbative QCD+EM [40]. Hence, it is consistent to assume that the ChPT LO of

qq

= 2B0F

20

+ does not introduce any residual e2 dependence when performing the

charge splitting in the low-energy expression. Using a different splitting prescription wouldlead in general to different bounds and a different definition of the EM LEC. For instance, an

alternative splitting procedure is introduced in [40] by matching running parameters of the e2

theory with those of the e2 = 0 one at a given matching scale 1. In that way, the EM part canbe chosen as 0-independent but it depends on the matching scale 1. We will not consider

that splitting here, since there are no available theoretical estimates for the LEC defined with

that scheme. The scale dependence for the LEC in either scheme is roughly expected to be

logarithmic.

Having the above considerations in mind, and going back to the case of physical quark

charges, we separate the EM corrections to the condensate through the ratio

qqe=0

qq

e=0 = 1 + 2

0 + e2Kr2() +O(p4)

= 1 + e2Kr2() 4Ce2

F40 +O(2 )+O(p4), (15)

with i defined in (13) and where we have expanded in M2 M20

/M2

0 0.1, as

0 = M20 (0/F2) +O(2 ), which is numerically reliable and can be performedin addition to the chiral expansion, in order to simplify the previous expression.

Wenotethatin SU(2) and tothisorder, the ratio (15) is notonly finite andscale independent

but it is also independent of the not-EM LEC, including the contact h1, h3, and therefore free

of ambiguities related to the condensate definition. In fact, this ratio is also independent of

B0, unlike the individual quark condensates, which only have physical meaning and give rise

to observables when multiplied by the appropriate quark masses, since miB0 M2i . In SU(2),the above ratio does not depend on the mass difference md mu either, i.e. it depends onlyon the sum

m and its deviations from unity are therefore purely of EM origin. All these

properties make the ratio (15) a suitable quantity to isolate the EM effects on the condensate.Thus, the ferromagnetic-like nature of the chiral order parameter qq, within its low-energyrepresentation, would require that this ratio is greater than 1, or equivalently to this order,

qq/e2 1. That condition leads to the following lower bound for the combination ofEM LEC involved to this order, neglecting the O(2 ) in (15) which changes very little the

numerical results:

5

kr5() + kr6()+ kr7 9C

F40 . (16)

We remark that the bound (16) is independent of the low-energy scale at which the

LEC on the left-hand side are evaluated as long as the same scale is used on the right-

hand side. Thus, it provides a well-defined low-energy prediction, expressed in terms of

meson masses. The LEC on the left-hand side could be estimated by fitting low-energy

processes or theoretically from the underlying theory, with all the related subtleties commentedabove.

Condition (16) and the corresponding ones for SU(3) that will be derived in section 4.4

are obtained as a necessary condition that the LEC should satisfy if the QCD physical vacuum

is ferromagnetic. This positivity condition on the quark condensate probes the vacuum by

taking the mass derivatives (9) through the external source method so that the quark masses

have to be kept different from zero and in that way the explicit symmetry-breaking corrections

are revealed in the condensate. If one is interested in the chiral limit, it must be taken only

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included are the following:

qqSU(3)l

uu + ddSU(3) = 2F2B0

1 + 8B0F2

m

2Lr8() +Hr2 ()

+ 4(2m + ms)Lr6()+ e2Kr3+() 13 3 sin2 0 2 K0K

1

3(1 + sin2 ) +O(p4)

(17)

uu ddSU(3) = 2F2B0

4B0

F2(md mu)

2Lr8() +Hr2 ()

e2Kr3()+ sin2

3

0

+ K K0

+O p2 (18)

ss

= F2B0 1 +

8B0

F2ms 2Lr

8

()+

Hr

2

()+ 4(2 m + ms)Lr

6

()+ e2K

r

s

()

43

0 sin

2 + cos2 2 [K + K0 ] +O p4

, (19)

where we use the notation (13) and

Kr3+() = 49

6

K7 + Kr8()+ 5 Kr9() + Kr10() ,

Kr3() = 43

Kr9() + Kr10()

, (20)

Krs() = 89

3

K7 + Kr8()+ Kr9() + Kr10() .

Note that in some of the above terms we have preferred, for simplicity, to leave the results

in terms of quark instead of meson masses. An important difference between the SU(2) and

SU(3) cases is that now there are loop corrections in uu dd, where eta and pion loops enterthrough the mixing angle and kaon ones through the chargedneutral kaon mass difference.

We have checked that the results are finite and scale independent with the renormalization

of the LEC given in the appendix and that they agree with [3] for e = 0. Some unpublishedresults related to the SU(2) and SU(3) isospin-breaking condensates can also be found in [50].

Numerical results for the condensates to this order can be found in [26]. As explained in

section 2.1, the effect of the Kri constants (20) in the condensates is not fully considered in that

work, where the EM contributions are included through the corrections of Dashens theorem

[34], so that only the Kri combinations entering mass renormalization appear. Then, we will

use our results with all corrections included to estimate the range of sensitivity to the Kri of

the condensates, analyzing the possible relevance for the fit in [26]. Our results are displayed

in table 1. As discussed in section 2.1, we take the same input values Lr6 = 0, ms/m = 24 asin [26] as well as the assumption H

r2 = 2L

r8, and the output values of B0mu,d,s,mu/md,F,L

r8

from their main fit. In the second and third columns of table 1, we give the results with

all the EM Kri fixed to their minimum and maximum natural values. Since the Kri appear

all with positive sign in (20), the absolute values of the condensates obtained in this way

are, respectively, lower and upper bounds within the natural range. We compare with the

results quoted in [26] to the same O(p4) order (fourth column) for their main fit and we

also show for comparison the results in the isospin limit e = 0, mu = md (fifth column).Our results agree reasonably with [26], although we note that the values in that work lie

12


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Table 1. Results for quark condensates. We compare with the values of [26] to O(p4) using thesame set of low-energy parameters as in the main fit of that work, except the Kri , which we considerat their lower (second column) and upper (third column) natural values. We also quote the valuesin the isospin limit to the same chiral order.

Kr

710 = 1

162 K

r

710 =1

162 [26] value O(p

4) Isospin limit

uu0/(B0F2) 1.278 1.292 1.271 1.290dd0/(B0F2) 1.297 1.305 1.284 1.290ss0/(B0F2) 1.899 1.907 1.964 1.904dduu 1 0.015 0.010 0.013 0

outside the natural range for the individual condensates. The largest relative corrections are

about 2% for the light condensates and about 4% for the strange one. These isospin-breaking

corrections are significant given the precision of the condensates quoted in [26]. On the other

hand, the corrections lie within the error range quoted by lattice analysis [31]. In turn, note

the bad ChPT convergence properties of the strange condensate, somehow expected sincess is much more sensitive to the strange quark mass ms than the light condensate [27] andtherefore the large strange explicit chiral symmetry breaking ms is responsible in this case

for the spoiling of the ChPT series, based on perturbative mass corrections. For the vacuum

asymmetry dduu 1, the natural value band covers the result in [26], although the numericaldiscrepancies in that case are relatively larger, between 15% and 24% for the lower and upper

limits of the Kri , respectively. Recall that this quantity vanishes to LO in ChPT, according to

(18), so that we expect it to be more sensitive to the Kri correction, which in this case comes

mostly from the combination Kr9 + Kr10. Nevertheless, it is worth noting that the results [26]imply dd/uu > 1 and ss/uu > 1, both in disagreement with many sum rule estimatesof the condensate ratios [29]. Not surprisingly, we have the same discrepancy, since we use

the same ChPT approach and the same numerical constants, except for the Kri corrections.

The discrepancy in the relatively large value of

ss

/

uu

comes possibly from the bad

convergence of theChPT series for thestrange condensate, which in addition is very sensitive tothe choice of Hr2 [26]. The light condensates converge much better and although the sign of

dd/uu 1 is under debate, its magnitude is very small. In the latter case, our presentcalculation may become useful since the Kr9 + Kr10 contribution may change the sign of thevacuum asymmetry, although its precise value to fit a given prediction for dd/uu 1would still be subject to the Hr2 value. For this reason, it is important to make predictions

for quantities which are independent of this ambiguity, as we have done in section 3 and as

we will do in section 4.2, where the sum rule for condensate ratios will allow us to make a

more reliable estimate of the vacuum asymmetry including both sources of isospin breaking.

Finally, we comment on the numerical differences by considering the more recent low-energy

fits in [28]. Still keeping Hr2 = 2Lr8, these new values for the low-energy parameters increaseconsiderably the total and strange condensates, which to O(p4) give qq/(2B0F2) 2.15and ss/(B0F

2

) = 2.79. These higher values are mostly due to the much smaller F =65 MeV, obtained in the main fit of [28] to accommodate a rather high Lr4 also with a large

error Lr4 = (0.75 0.75) 103 (an output result in [28]). With the previous value F = 87.1MeV but keeping the rest of LEC and masses as in [28] we obtain qq/(2B0F2) 1.63 andss/(B0F2) = 1.99. The EM corrections remain of the same size and therefore theirrelative effect is somewhat smaller. As commented before, mu = md isospin breakingis not implemented in those new fits and EM corrections are included only in kaon

masses.

13


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4.2. Sum rule corrections

As noted in [3], for mu = md one can combine the isospin-breaking condensates into a sumrule relating the isospin asymmetry dd/uu with the strange one ss/uu. Such relation is

phenomenologically interesting because it does not include contact terms and hence is suitablefor numerical estimates on the size of the isospin-breaking corrections. Our purpose in this

section is to discuss the EM e = 0 contribution to that sum rule. To LO in mu md and e2 wefind

SR dduu 1 +

md mums m

1 ssuu

= mu mdms m

1

162F2

M2K M2 +M2 log

M2

M2K

+ e2

C

82F4

1 + log M

2K

2

8

3

Kr9() + Kr10()

. (21)

The last term proportional to e2 is scale independent and is the charge correction to the

result in [3]. With the numerical set we havebeenusing, the mdmu term on the right-hand sidegives 3.3103, whereas the e2 term gives 3.37103 with Kr9(M )+Kr10(M ) = 1/(82)and 9.4 104 with Kr9(M ) + Kr10(M ) = 2.7 103, the central value given in [34].Therefore, the charge term above is of the same order as the pure QCD isospin correction and

must be included when estimating the relative size of condensates through this sum rule. In

fact, using the values quoted in [29] mu/md = 0.55, ms/md = 18.9 and ss/uu = 0.66, weobtain from (21) with physical pion and kaon masses

0.015 < dduu 1 < 0.009,where the lower (upper) bound corresponds to the natural value Kr9 + Kr10 = +()1/(82),whilethe value without considering the charge correctionis 0.012 and the value quotedin [29]collecting various estimates in the literature is 0.009. The inclusion of the charge correctionsmay then help to reconcile this sum rule with the different condensate estimates available.In fact, through this sum rule we see that ChPT is also compatible with the asymmetries

dd/uu and ss/uu both being smaller than 1 (see our comments in section 4.1). Note thatthe ferromagnetic-like arguments used in sections 3 and 4.4 cannot be applied to uu dd,which does not behave as an order parameter under chiral transformations, since it is not

invariant under SUV(2). Finally, we recall that estimates based on the sum rule (21) are more

precise than the ones we have made directly from the condensates in section 4.1, since this

sum rule is free of the Hr2 ambiguity.

4.3. Matching of LEC

Our aim in this section is to explore the consequences of including the two sources of

isospin breaking for the matching of the LEC involved in the condensates. For that purpose,

we perform a 1/ms expansion in the SU(3) sum and difference condensates given in(17)(18). Matching theO(1) andO(log ms) terms with the corresponding SU(2) expressions

in (10)(11) yields the following relations between the LEC, for the sum and difference of

condensates respectively:

2M20

lr3() + hr1()+ e2F2Kr2() = 2M20 16Lr6() + 4Lr8() + 2Hr2 () 18 K02

+ e2F2

K

r3+()

2C

F4K0

, (22)

14


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B0(md mu)h3 2e2F2

3k7 = B0(md mu)

4Lr8() + 2Hr2 ()

3 K0

2+ 1

962

2e2F2

3Kr9() + K

r10()

3C

2F4K0 . (23)

In the above expressions, we have displayed the SU(2) contribution on the left-hand side

and the SU(3) ones on the right-hand side, with Kr2() and Kr3+() given in (12) and (20).

Note that the 1/ms expansion has been implemented also in the tree-level relations (7), so

that M20

= (mu + md)B0 + O(1/ms), M2K0 = B0ms + O(1) and M2 = 4B0ms/3 +O(1). Itis important to point out that the pion mass charge difference is not negligible in the 1 /msexpansion, and for that reason we keep M0 in (22). For kaons, it is justified to consider the

charge contribution negligible against the dominant ms term, so that at this order MK and MK0

are not distinguishable.

In the sum matching relation (22), the isospin corrections are not very significant. The

mass difference mu md does not appear in the neutral and kaon masses to LO in 1/msand the charge correction, although of the same chiral order as the M2

0term, numerically

e2F2/M20

Ce2/(F2M20) 0.05. However, in the difference matching (23), the mu md

corrections contribute on the same footing as the EM ones and are numerically comparable.

The above matching relations can be used directly for the approximated LEC (estimated

theoretically or fitted to data) and for physical masses, since the difference from the tree-level

masses and LEC is hidden in higher orders. On the other hand, for the tree-level LEC, i.e. the

ChPT O(p4) Lagrangian parameters, since they are formally independent of the light quark

masses, we can just take the chiral limit mu = md = 0 in the above expressions (22)(23)and read off the corresponding matching of the e2 contributions. Using the latter again in

(22)(23) then gives independent relations between the tree-level LEC involved at e2 = 0 andthe EM ones. Doing so, the EM and not-EM LEC combinations decouple and the results are

compatible with those obtained in [3] for e = 0 and in [43] for e = 0 (setting mu = md = 0from the very beginning):

lr3() + hr1() = 16Lr6() + 4Lr8() + 2Hr2 () 18 K0

2,

h3 = 4Lr8() + 2Hr2 ()

3 K0

2+ 1

962,

5(kr5() + kr6()) = 6(K7 + Kr8()) + 4(Kr9() + Kr10())3C

F4K0,

k7 = Kr9() + Kr10() 3C

2F4K0 ,

(24)

where the functions are evaluated exactly in the chiral limit, i.e. for M2K0

= B0ms andM2 = 4B0ms/3, the first and third equations coming from (22) and the second and fourthfrom (23).

Then, our first conclusion is that to this order of approximation, the formal matching of

the condensates is consistent with the matching relations previously obtained. In other words,mass and charge terms can be separately matched. This would be no longer true at higher

orders where for instance e2(mu md) contributions may appear.Although relations (22) and (23) reduce to (24) in the chiral limit for the tree-level LEC, it

is better justified to use the original expressions (22)(23) when dealing with physical meson

masses and when the LEC are obtained either from phenomenological or theoretical analysis.

The LEC obtained in that way are approximations to the Lagrangian values and consequently

they depend on mass scales characteristic of the approximation method used. For instance, the

15


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LEC obtained by phenomenological fits are sensitive to variations both in m and in mu md[26], in resonance saturation approaches they depend on vector meson masses [37, 38] which

themselves depend on quark masses and in the NJL model some LEC such as Kr10 depend on

the scale where the quark masses are renormalized [34]. We do not expect large differences

between using the general matching relation (22) or the first and third equations in (24), sincethe latter can also be understood as the e = 0 limit of the former and we have seen thatthis is numerically a good approximation. However, that is not so clear for (22) where the

two isospin-breaking contributions are of the same order, both in the chiral expansion and

numerically.

Finally, we can use the previous matching relations to estimate numerically the SU(2)

condensates in (10)(11) without having to appeal to the values of the SU(2) LEC. Doing so

we obtain uu + ddSU(2)/B0F2 (2.16, 2.18) and uu ddSU(2)/B0F2 (0.014, 0.02)where we indicate in brackets the natural range of the EM LEC, to be compared to

uu + ddSU(3)/B0F2 (2.58, 2.6) and uu ddSU(3)/B0F2 (0.013, 0.018) fromtable 1. The larger difference in uu + dd comes from the O(ms) and O(ms log ms) termsin the 1/ms expansion, which were separated when doing the matching and which are

absent in the condensate difference. In fact, the numerical contribution of those terms touu + ddSU(3)/B0F2 is about 0.41, which explains perfectly the numerical differences andconfirms the idea that in standard ChPT the light condensates calculated either in the SU(2)

or in the SU(3) cases give almost the same answer near the chiral limit. This may be not the

case in other scenarios of chiral symmetry breaking [32].

4.4. EM corrections and SU(3) LEC bounds

We have seen in the SU(2) case that the EM ratio given in (15) is a relevant physical quantity

allowing us to establish a constraint for the EM LEC based on explicit chiral symmetry

breaking. The same argument applied to the SU(3) case also leads to a constraint on the

EM LEC obtained from the full condensate qq = uu + dd + ss, which behaves as anorder parameter, being an isosinglet under SUV(3). In addition, we can still consider the light

condensate qql as the order parameter of chiral transformations of the SU(2) subgroup,which in principle will lead to a different constraint. In fact, the latter is nothing but the

constraint obtained in the SU(2) case (16), once the equivalence between the LEC obtained

in section 4.3 is used. As for the full condensate, it should be kept in mind that the large

violations of chiral symmetry due to the strange quark mass may spoil our simple description

of small explicit breaking. As commented above, this reflects to the strange condensate in

the large NLO contributions, which in the standard ChPT framework depends strongly on ms,

unlike the light condensate. Therefore, the bounds of the LEC obtained for the full condensate

are less trustable, since neglecting higher orders, say ofO(e2ms), is not so well justified for

ss. Proceeding then as in section 3, where the same prescription of charge splitting whencomparing with QCD approaches is understood, we calculate the ratios

qqe=0l

qqe=0l

SU(3) = 1 + 4e2

9

6

K7 + Kr

8()+ 5 Kr9() + Kr10() 2Ce

2

F4 [2 + K ]+O(2 , 2K)+O(p4) (25)

qqe=0qqe=0

SU(3)= 1 + 8

9e2

2(Kr9() + Kr10())+ 3(K7 + Kr8()) 8Ce2

3F4[ + K ]

+O(2 , 2K)+O(p4), (26)16


18/22


qq l qq

1.0 0.5 0.0 0.5 1.01.0

0.5

0.0

0.5

1.0

82 K7 K8

r

8

2

K9

r

K10

r

Figure 1. Regions in the LEC space constrained by the bounds on the light condensate (27), abovethe full blue line, and the full one (28), above the dashed red line. The LEC are renormalized at

=M and are plotted within the natural range.

where, asin the SU(2) case, we have expanded in and alsoin K = (M2K M2K,e=0)/M2K 0.008, which allows us to express the results in terms of the full and K masses. Otherwisewe should take into account that now M(e = 0) = M0 , unlike the SU(2) case, and

MK (e = 0) = MK0 , by terms of order md mu. This is important when using this result fornumerical estimates, since, as discussed before, the separation of the e = 0 correction to themasses is formally not unique. As in SU(2), the ratios (25)(26) are finite and independent of

the scale , of B0 and of the not-EM LEC, so they are free of contact ambiguities.

As in section 3, we want to explore the consequences of the ferromagnetic nature of

the physical QCD vacuum under explicit chiral symmetry breaking for the EM LEC. Here,

also the charge-mass crossed terms in the fourth-order Lagrangian (A.2) give explicit breaking

contributions to the quarkcondensatecoming now from the isoscalar, isovectorand strangenesspart of the charge matrix. For physical quark charges, demanding that the ratios ( 25)(26) are

greater than 1 we obtain the following EM bounds, to LO in the chiral expansion and in , K:

qql 6

K7 + Kr8()+ 5 Kr9() + Kr10() 9C2F4 (2 + K ) (27)

qq 2 K7 + Kr8()+ 3 Kr9() + Kr10() 3CF4

( + K ) . (28)

We remark that these constraints are independent of the low-energy scale . It is also

clear that the light bound (27) is nothing but the one obtained in the SU(2) case (16) once the

equivalence between the LEC given in the third equation of (24) is used. In figure 1, we have

plotted these two constraints in the (K7 + Kr8 ) (K

r9 + K

r10) plane at = M and within the

natural region. We have used the same numerical values for the tree-level LO masses Cand F

as in previous sections. We observe that the bound on the full condensate is more restrictive

than the light one in that range. However, as we have commented above, it is also less trustable,

due to the large distortion of the chiral invariant vacuum due to the strange mass. Both bounds

also give a more restrictive condition than just the natural size.

Let us now check these bounds against some estimates of the Kri in the literature. We

start with the sum rule approach for Kr7...10 in [37]. In that work, K7 = Kr8(M ) = 0, but what17


19/22


is more relevant for us is that the combination Kr9 + Kr10 at any scale is gauge independentdespite being both Kr9, K

r10 dependent on the gauge parameter , as one can readily check from

the explicit expressions given in [37] (cf their equations (94) and (95)). This is an interesting

consistency check of our present bounds (27) and (28), which are gauge independent in

addition to their low-energy scale independence commented previously, supporting theirvalidity and predictive character. Numerically, the constant Kr9 could not be estimated in [37]

due to the slow convergence of the integrals involved, but they provide a numerical estimate for

Kr10(M ) = 5.2 103 at 0 = 0.7 GeV and = 1, for which we obtain Kr9(M ) 0.021from (27) and Kr9(M ) 0.015 from (28). See our comments about the0 scale dependencein section 3.

In [35] resonance saturation gives Kr7...10(M ) = 0, which is compatible with our presentbound. This is apparently incompatible with a previous value for Kr8(M ) = (41.7)103obtained in [4]. The possible reasons to explain this difference were discussed in [35]. That

value for Kr8 is compatible with our bounds as long as 6K7 + 5(Kr9 + Kr10) 0.05 from (27)and 2K7 + 3(Kr9 + Kr10) 0.02 from (28).

In [34], based on large Nc and the NJL model, the LEC estimates give K7 = 0,K8(M ) = (0.82.0)10

3

(K7 and K8 areO(1/Nc) suppressed) and K

r

9(M )+Kr

10(M ) =(2.7 1.0) 103, all of them at 0 = 0.7 GeV. These values are also compatible with bothbounds (27)a nd(28). We notethat in[36], where the short-distance contributions are evaluated

as in [34], the explicit expressions given for the LEC again show that K7,Kr8 and K

r9 + Kr10 are

gauge independent. Furthermore, Kr9 and Kr10, dominant for large Nc in that approach, show a

rather large stability range in the 0 scale around 0 = 0.7 GeV [34, 36]. Since Kr9 + Kr10 isthe only combination surviving for large Nc in our bounds, the comparison with those works

is robust concerning the gauge and QCD scale dependence.

5. Conclusions

In this work, we have carried out an analysis of strong and electromagnetic isospin-breaking

corrections to the quark condensates in standard one-loop ChPT, providing their explicit

expressions and studying some of their main phenomenological consequences for two andthree light flavours.

Our results have allowed us to analyze the sensitivity of recent isospin-breaking numerical

analysis of the condensates to considering all the EM LEC involved. The effect of those LEC

is smaller for individual condensates than for the vacuum asymmetry, where they show up

already in the LO. These corrections lie within the error range quoted in lattice analysis. Our

analysis can also be used to estimate corrections to the quark condensate coming from lattice

artificially large meson masses.

We have shown that if EM explicit chiral symmetry breaking induces a ferromagnetic-

like response of the physical QCD vacuum, as in the case of quark masses, one obtains useful

constraints as lower bounds for certain combinations of the EM LEC, both in the two and three

flavour sectors. We have explored the consequences of this behaviour for the ratios ofe = 0 toe = 0 light and total quark condensates, which are free of contact-term ambiguities, and for agiven convention of charge separation. The large ChPT corrections to the strange condensate

make the constraints on the full condensate less reliable. In this context, we have discussed

the different sources for EM explicit chiral symmetry-breaking and isospin-breaking terms, by

considering formally arbitrary quark charges. Thus, there are chiral symmetry-breaking terms

proportional to the sum of charges squared, coming from crossed charge-mass contributions

in the effective action, which show up in the vacuum expectation value. In accordance with

the external source method, we keep the quark masses different from zero to account correctly

18


20/22


for all the explicit symmetry-breaking sources. The chiral limit can be taken at the end of the

calculation. The bounds obtained are explicitly independent of the low-energy scale , then

providing a complete and model-independent prediction at low energies. However, when this

low-energy representation is compared with theoretical estimates based on QCD, one has to

take into account that due to the convention used in the charge separation, the estimated LECdepend on the QCD renormalization scale 0, as well as being gauge dependent. Our bounds

are numerically compatible with those estimates, based on sum rules, resonance saturation

and QCD-like models, within the stability range of 0 where those approaches are reliable.

Furthermore, the LEC combinations appearing in our bounds are gauge independent. We

believe that our results can be useful in view of the few estimates of the EM LEC in the

literature.

We have found that the EM correction to the sum rule relating condensate ratios is of the

same order as the previously calculated e = 0 one, and therefore must be taken into accountwhen using this sum rule to estimate the relative size of quark condensates. We have actually

showed that using the complete sum rule, which is also free of contact terms, yields a ChPT

model-independent prediction for the vacuum asymmetry compatible with the results quoted

in the literature.Finally, we have performed a matching between the SU(2) and SU(3) condensates,

including all isospin-breaking terms. Matching the sum and difference of light condensates

gives rise to matching relations between the LEC involved, where EM and not-EM LEC enter

on the same footing in the chiral expansion. These matching relations may be useful when

working with physical masses and LEC estimated by different approximation methods. In the

case of the sum, the charge contribution is numerically small with respect to the pion mass

one, but in the difference the two sources of isospin breaking are comparable. Taking the chiral

limit, EM and not-EM constants decouple and the matching conditions are compatible with

previous works for the LEC in the Lagrangian, which are defined in this limit.

Acknowledgments

We are grateful to J R Pelaez and E Ruiz Arriola for useful comments. RTA would like to

thank Buenaventura Andres Lopez for invaluable advice. This work was partially supported

by the Spanish research contracts FPA2008-00592, FIS2008-01323, UCM-Santander 910309

GR58/08, GR35/10-A and the FPI programme (BES-2009-013672).

Appendix. Fourth-order Lagrangians and renormalization of the LEC

Here, we collect some results available in the literature and needed in the main text. To

calculate the quark condensates to NLO one needs the Lp4+p2e2+e4 Lagrangians to absorb thedivergences coming from loops with vertices from Lp2+e2 . We denote by a superscript qq therelevant terms in the Lagrangian, which are those containing the quark mass matrix. For SU(2)

they are [5, 6]

Lqq

p4= l3

16tr[(U + U)]2 + 1

4(h1 + h3)tr[2] +

1

2(h1 h3) det(),

Lqq

p2e2= F2(k5tr[(U+ U)]tr[Q2] + k6tr[(U+ U)]tr[QUQU] + k7tr[(U + U)Q

+ (U + U)Q]tr[Q]), (A.1)

19


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where = 2B0M, whereas for SU(3) [4]L

qq

p4= L6tr[(U+ U)]2 +L8tr[UU+ UU] +H2tr[2],

Lqq

p2e2= F2(K7tr[(U+ U)]tr[Q2] + K8tr[(U+ U)]tr[QUQU]

+ K9tr[(U + U + U + U)Q2] + K10tr[(U+ U)QUQU+ (U + U)QUQU]), (A.2)

and Lqq

e4= 0 for both cases.

In order to renormalize the meson loops it is necessary to separate the LECs appearing in

the NNLO Lagrangian into finite and divergent parts. The renormalization of the LEC involved

in the calculation of the SU(2) condensates is given by [2, 5, 6]

li = lri () + i,hi = hri () + i,ki = kri () + i,

with 3

= 12

, 1

=2, 3

=0, and 5

= 14

15Z, 6

=14

+2Zand 7

=0, for physical quark

charges eu = 2e/3, ed = e/3, being Z := CF4 . The part that diverges in d = 4 dimensions isisolated from the loop integrals and is expressed as

= d4

162

1

d 4 1

2[log 4 + (1)+ 1]

,

where (1) is the Euler constant. As for the SU(3) ones, we have [3, 4]Li = Lri () + i,Hi = Hri () +i,Ki = Kri () + i,

with 6 = 11144 , 8 = 548 , 2 = 524 , and 7 = 0, 8 = Z, 9 = 14 , 10 = 14 + 32Z.

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